Here \(D(u)\)  is the data term and \(R(u)\)  is the regularization term.
Instead of minimizing the original functional (REF ), it is convenient to minimizing (REF ) with the auxiliary variable \(y\) . In discrete optimization including the binary optimization, the auxiliary variable \(y\)  is called “line process" in [1]}, [2]}, [3]} or more general “outlier process" [4]}. These line or outlier processes are popular and widely used, not only because they have physical or statistic intuition and the ability to model spatial properties of discontinuities [4]} but also they can make the optimization procedure more efficient and stable. Actually, one can get (REF ) from the original (REF ) by the Fenchel-Rockafellar duality theory. We refer to the Geman-Reynolds model [2]} and Geman-Yang model [3]} which are two typical and different duality processes to get (REF ) from (REF ). In [8]}, the alternating minimization focused on convex regularization term \(R(u)\)  is discussed.
In [9]}, [10]}, [11]} and the thesis [12]}, two alternating minimization algorithms including “ARTUR" and “LEGEND" algorithms were developed with global convergence analysis for convex models with convex regularizations.
Although there are rich studies on alternating minimization for the half quadratic models or the Ambrosio-Tortorelli model, however, solving the large scale linear subproblems for \(u\)  (or \(y\) ) is still very challenging. Recent attempts can be found in [13]}, [14]} where conjugate gradient method with line search as an inexact solver was developed. Based on the weighted quadratic approximations on \(F(u)\) , truncated conjugate gradient method with error control was also developed for \(C^1\)  function \(F(u)\)  in [15]}.
